Diameter two properties and almost square properties in Lipschitz-free spaces and spaces of Lipschitz functions
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Tartu Ülikooli Kirjastus
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Diameeter-2 omadused ja peaaegu ruudu omadused kirjeldavad Banachi ruumi ühikkera kuju. Täpsemalt näitavad need omadused, kas kõik ühikkera teatud osahulgad on maksimaalse võimaliku diameetriga. Need omadused moodustavad Banachi ruumide geomeetria olulise uurimissuuna.
Doktoritöös uuritakse diameeter-2 omadusi ja peaaegu ruudu omadusi Lipschitzi funktsiooniruumides ning nende loomulikes eelruumides, Lipschitzi-vabades ruumides. Töös tõestatakse, et Lipschitzi-vaba ruum üle liinkaugusega meetrilise ruumi on lokaalse peaaegu ruudu omadusega. Teisalt näidatakse, et ühelgi Lipschitzi-vabal ruumil ei saa olla peaaegu ruudu omadust. Lisaks konstrueeritakse lokaalse peaaegu ruudu omadusega Lipschitzi-vaba ruum, millel ei ole nõrka peaaegu ruudu omadust. Väitekirjas uuritakse ka diameeter-2 omadusi Lipschitzi funktsiooniruumides ja esitatakse neile uued meetrilised kirjeldused. Uute näidetega eristatakse Lipschitzi funktsiooniruumides kõik diameeter-2 omadused
Diameter two properties and almost square properties describe the geometric shape of the unit ball of a Banach space. In particular, they express whether certain subsets of the unit ball necessarily have the maximal possible diameter. These properties constitute an important direction in the study of the geometry of Banach spaces. This thesis investigates diameter two properties and almost square properties in spaces of Lipschitz functions and in their canonical preduals, the Lipschitz-free spaces. It is shown that a Lipschitz-free space over a length metric space is locally almost square. On the other hand, it is proven that no Lipschitz-free space is almost square. Moreover, an example of a Lipschitz-free space that is locally almost square but not weakly almost square is constructed. The thesis also studies diameter two properties in spaces of Lipschitz functions and provides new metric characterisations of these properties. Using new examples, all diameter two properties are shown to differ in these spaces.
Diameter two properties and almost square properties describe the geometric shape of the unit ball of a Banach space. In particular, they express whether certain subsets of the unit ball necessarily have the maximal possible diameter. These properties constitute an important direction in the study of the geometry of Banach spaces. This thesis investigates diameter two properties and almost square properties in spaces of Lipschitz functions and in their canonical preduals, the Lipschitz-free spaces. It is shown that a Lipschitz-free space over a length metric space is locally almost square. On the other hand, it is proven that no Lipschitz-free space is almost square. Moreover, an example of a Lipschitz-free space that is locally almost square but not weakly almost square is constructed. The thesis also studies diameter two properties in spaces of Lipschitz functions and provides new metric characterisations of these properties. Using new examples, all diameter two properties are shown to differ in these spaces.
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